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Subadditivity of Syzygies of Ideals and Related Problems SUBADDITIVITY OF SYZYGIES OF IDEALS AND RELATED PROBLEMS JASON MCCULLOUGH Decidated to Professor David Eisenbud on the occasion of his seventy-fifth birthday Abstract. In this paper we survey what is known about the maximal degrees of minimal syzygies of graded ideals over polynomial rings. Sub- additivity is one such property that is conjectured to hold for certain classes of rings but fails in general. We discuss bounds on degrees of syzygies and regularity given partial information about the beginning of a free resolution, such as degrees of generators or degrees of first syzy- gies. We also focus specifically on conditions that guarantee an ideal is quadratic with linear resolution for a fixed number of steps. Finally we collect some old and new open problems on degrees of syzygies. 1. Introduction Graded free resolutions are highly useful vehicles for computing invariants of ideals and modules. Even when restricting to finite minimal graded free resolutions of graded ideals and modules over a polynomial ring, there are questions regarding the structure of such resolutions that we do not yet understand. The aim of this survey paper is to collect known results on the degrees of syzygies of graded ideals and pose some open questions. Let K be a field and let S = K[x1; : : : ; xn] denote a standard graded polynomial ring over K. If M is a finitely generated, graded S-module, let S βi;j(M) = dimk Tori (M; k)j denote the graded Betti numbers of M, and let ti(M) = supfj j βi;j(M) 6= 0g denote the ith maximal graded shift of M. Thus ti(M) denotes the maximal degree of an element in a minimal generating set of the ith syzygies of M. The shifts ti(M) are primarily of interest due to their connection with another invariant, the regularity reg(M) of M; indeed one can take reg(M) = maxfti(M) − ig as a definition of the regularity of M. The underlying question considered in this paper is the following: Question 1.1. Which sequences of integers (t0; t1;:::; tm) can be realized as (t0(S=I); t1(S=I);:::; tm(S=I)) for some graded ideal I ⊆ S? Note that the more general question in which S=I is replaced by an ar- bitrary graded module M is not so interesting. One of the main results of Key words and phrases. free resolution, regularity, polynomial ring, projective dimension. 1 2 J. MCCULLOUGH Boij-S¨oderberg theory, proved by Eisenbud, Fløystad, and Weyman [23] in characteristic 0 and Eisenbud and Schreyer [26] in all characteristics, shows that every strictly increasing sequence t0 < t1 < ··· < tc of integers with c ≤ n can be realized as the shifts of some graded, Cohen-Macaulay, pure S-module of codimension c. On the other hand, one sees immediately that the same statement is not true for cyclic modules S=I. Indeed, the sequence (0; 1; 3; 4) cannot be realized as the maximal graded shifts of any cyclic module S=I. If it could, then I would be generated by linear forms (since t1(S=I) = 1) and yet would have minimal quadratic syzygies, which is im- possible; any such S=I would be resolved by a Koszul complex with linear differential maps. This degree sequence (0; 1; 3; 4) can be realized by the res- olution of Coker(M), where M is a generic 2 × 4 matrix. (See Example 2.1.) This explains restricting our attention to ideals and cyclic modules. After a section to collect notation and background, there are four main sections to this paper, each dealing with refinements to Question 1.1. In Section 3, we consider effective bounds on reg(S=I) in terms of some initial segment of the maximal shifts. Thought of another way, we address the question of how much of the resolution of an ideal must be computed to get a reasonable bound on its regularity. In Section 4, we consider the subadditivity property of maximal graded shifts; that is, when ta(S=I) + tb(S=I) ≥ ta+b(S=I) for all a; b ≥ 1. It is not hard to find examples where this property fails, but for specific classes of ideals, subadditivity has been proved or conjectured. In Section 5, we consider bounds on maximal graded shifts for arbitrary ideals. In Section 6, we focus specifically on ideals generated by quadrics with linear resolutions for a fixed number of steps. We examine geometric and combinatorial conditions which guarantee resolutions of this form. In the final Section 7, we collect some open questions and pose some new problems that we hope will inspire future work in the area. 2. Background In this section we fix notation used throughout this paper. Let K denote a field and let S = K[x1; : : : ; xn] denote a polynomial ring over K. We assume throughout that S is standard graded, i.e. deg(xi) = 1 for 1 ≤ i ≤ n. We write Si for the K-vector space spanned by all degree i homogeneous L polynomials so that S = i≥0 Si as K-vector spaces. We write S(−j) for a rank one free module with generator in degree j so that S(−j)i = Si−j. We consider the resolutions of graded ideals I = (f1; : : : ; fm) and graded modules M of S. Note that I is graded if it has a set of homogeneous generators. We write F• for the minimal graded free resolution of M so that L βij (M) Fi = j S(−j) . The numbers βij(M) are the graded Betti numbers S of M and can alternatively be computed as βij = dimk Tori (M; k)j. (We drop the module M from the notation when it is clear from context.) In particular, the graded Betti numbers are isomorphism invariants of M. It is convenient to keep track of the graded Betti numbers in a matrix called SUBADDITIVITY OF SYZYGIES 3 the Betti table; by convention, we place βi;i+j(M) in position (i; j). One of the advantages of the Betti table is that it allows us to quickly read off the projective dimension and regularity of the module being resolved. More precisely, we may define reg(M) = maxfj − i j βij(M) 6= 0g and pd(M) = maxfi j βi;j(M) 6= 0g. Thus reg(M) is the index of the last nonzero row in the Betti table of M and pd(M) is the index of the last nonzero column. If we want to study the structure of minimal, graded free resolutions more closely, we can consider the maximal and minimal graded shifts of M. For each i ≥ 0, set ti(M) = supfj j βij(M) 6= 0g and ti(M) = inffj j βij(M) 6= 0g: In other words, ti(M)(ti(M)) denotes the maximal (resp. minimal) degree of an element in a minimal generating set of the ith syzygy module of M. When M = S=I is a cyclic module, t1(S=I) denotes the maximal degree of a minimal generator of I. A module M is called pure if ti(M) = ti(M) for all 0 ≤ i ≤ pd(M). Example 2.1. Let S = K[x1; : : : ; x8] and let M = Coker(A), where x x x x A = 1 2 3 4 : x5 x6 x7 x8 is a generic matrix. M is resolved by a Buchsbaum-Rim complex [21, Ap- pendix A2.6] with the following Betti table. 0 1 2 3 0: 2 4 -- 1: -- 4 2 In particular, M is a pure module with t0(M) = t0(M) = 0, t1(M) = t1(M) = 1, t2(M) = t2(M) = 3, and t3(M) = t3(M) = 4. This is the example mentioned in the introduction. In a minimal graded free resolution, it is clear that the minimal graded shifts are strictly increasing, that is ti−1(S=I) < ti(S=I) for all 1 ≤ i ≤ pd(S=I). The maximal graded shifts are strictly increasing up to ht(I). Proposition 2.2 ([50, Proposition 2.2], [4, Lemma 5.1]). Let I be a standard graded ideal in a polynomial ring S = K[x1; : : : ; xn] over a field K. For all 1 ≤ i ≤ ht(I), one has ti−1(S=I) < ti(S=I): To see that ti−1(S=I) ≥ ti(S=I) is possible for i > ht(I), see Example 3.11. For the remainder of the paper, we primarily focus on the maximal graded shifts ti(S=I) for graded cyclic modules S=I. 4 J. MCCULLOUGH 3. Effective Bounds on Regularity In this section we consider bounds on the regularity of ideals in terms of some initial segment of the maximal graded shifts. Fix a graded ideal I and write d(I) for the maximal degree of an element in a minimal generating set of I. We recall that reg(S=I) = maxfti(S=I) − ig and so in particular reg(S=I) ≥ d(I) − 1 = t1(S=I) − 1. A natural question is to what extent reg(S=I) can exceed d(I). Without referencing the number of variables, no such bound is possible - an ideal I generated by a complete intersection of n quadrics has d(I) = 2 and reg(S=I) = n. If we fix the number of variables to be n, then there is a well-known doubly exponential upper bound on regularity, due to Bayer and Mumford in characteristic 0, and later Caviglia and Sbarra in all characteristics. Theorem 3.1 ([7, Proposition 3.8], [14, Corollary 2.7]). Let I be a graded ideal in S = K[x1; : : : ; xn].
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